The term “Ising system” may be used herein to denote any physical system, particularly a two-dimensional electron gas (2DEG) whose low energy effective theory is governed by the Ising topological quantum field theory (TQFT). Below, the defining characteristics of the Ising TQFT are set out in tabular form using the notation and formalism of modular tensor categories (MTC). Although the structure appears to require much data, actually this direct description of properties is quite redundant: according to Kitaev, there are precisely eight theories with the Ising fusion rules and these can be distinguished by a Chern class.
A microscopic construction of the Ising system was given in Kitaev [arXiv:cond-mat/050506438v3], incorporated herein by reference, and realization in optical lattices discussed in Duan, et al. [Phys. Rev. Lett., 91(9):090402, August 2003], incorporated herein by reference. The original microscopics was defined on a honeycomb lattice, but the only essential feature is that the bonds can be grouped into three distinct classes—x, y, and z—so that, at all vertices, exactly one bond from each class is present. This weaker condition allows the construction of such lattices on surfaces of any genus.
Another road to the Ising TQFT is through px+ipy superconducting 2DEGs. Such systems are predicted, based on elementary band theory, to arise in a variety of 2D-systems from a spin-orbit coupled semiconductor with superconductivity imported via proximity effect. Examples include Sau, et al. [arXiv:cond-mat/0907.2239v3], Alicea [arXiv:cond-mat/0912.2115v1], and Qi, et al. [arXiv:cond-mat/1003.5448v1], each of which is incorporated herein by reference. Such systems are chiral topological superconductors and support localized Majorana states.
These systems are not purely topological, but, being superconductors, also support a classical order parameter φ If the system is not planar, but configured as a surface of genus >0, a significant stiffness term λ|▾φ|2 in the Lagrangian may prevent superposition of certain topological states, which are correlated with the winding of φ, a classical quantity for extensive superconductors.
Within the Ising TQFT, braid operations, together with nondemolition measurement of the collective charge of up to four σ-particles, supports the implementation of all Clifford operations. It is known that if the “π/8-gate,”
                                                      1            〉                                                                        ϕ            〉                                ⁢                                                            ɛ                              ⅈπ                /                8                                                          0                                                0                                              ɛ                                                -                  ⅈπ                                /                8                                                                ,is added to the Clifford operations, a computationally universal gate results.